↑

Properties

Label 400.8.c.j
Level $400$
Weight $8$
Character orbit 400.c
Analytic conductor $124.954$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,8,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \beta q^{3} - 508 \beta q^{7} + 2043 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 6 \beta q^{3} - 508 \beta q^{7} + 2043 q^{9} - 1092 q^{11} - 691 \beta q^{13} + 7353 \beta q^{17} - 39940 q^{19} + 12192 q^{21} + 34356 \beta q^{23} + 25380 \beta q^{27} + 102570 q^{29} - 227552 q^{31} - 6552 \beta q^{33} + 80263 \beta q^{37} + 16584 q^{39} + 10842 q^{41} - 315374 \beta q^{43} - 236328 \beta q^{47} - 208713 q^{49} - 176472 q^{51} + 747009 \beta q^{53} - 239640 \beta q^{57} + 2640660 q^{59} + 827702 q^{61} - 1037844 \beta q^{63} + 63002 \beta q^{67} - 824544 q^{69} + 1414728 q^{71} - 490141 \beta q^{73} + 554736 \beta q^{77} - 3566800 q^{79} + 3858921 q^{81} + 2836446 \beta q^{83} + 615420 \beta q^{87} + 11951190 q^{89} - 1404112 q^{91} - 1365312 \beta q^{93} + 4341073 \beta q^{97} - 2230956 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4086 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4086 q^{9} - 2184 q^{11} - 79880 q^{19} + 24384 q^{21} + 205140 q^{29} - 455104 q^{31} + 33168 q^{39} + 21684 q^{41} - 417426 q^{49} - 352944 q^{51} + 5281320 q^{59} + 1655404 q^{61} - 1649088 q^{69} + 2829456 q^{71} - 7133600 q^{79} + 7717842 q^{81} + 23902380 q^{89} - 2808224 q^{91} - 4461912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 1.00000i
1.00000i
0 − 12.0000i 0 0 0 1016.00i 0 2043.00 0
49.2 0 12.0000i 0 0 0 − 1016.00i 0 2043.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.8.c.j 2
4.b odd 2 1 50.8.b.c 2
5.b even 2 1 inner 400.8.c.j 2
5.c odd 4 1 16.8.a.b 1
5.c odd 4 1 400.8.a.l 1
12.b even 2 1 450.8.c.g 2
15.e even 4 1 144.8.a.i 1
20.d odd 2 1 50.8.b.c 2
20.e even 4 1 2.8.a.a ✓ 1
20.e even 4 1 50.8.a.g 1
40.i odd 4 1 64.8.a.e 1
40.k even 4 1 64.8.a.c 1
60.h even 2 1 450.8.c.g 2
60.l odd 4 1 18.8.a.b 1
60.l odd 4 1 450.8.a.c 1
80.i odd 4 1 256.8.b.f 2
80.j even 4 1 256.8.b.b 2
80.s even 4 1 256.8.b.b 2
80.t odd 4 1 256.8.b.f 2
120.q odd 4 1 576.8.a.g 1
120.w even 4 1 576.8.a.f 1
140.j odd 4 1 98.8.a.a 1
140.w even 12 2 98.8.c.d 2
140.x odd 12 2 98.8.c.e 2
180.v odd 12 2 162.8.c.a 2
180.x even 12 2 162.8.c.l 2
220.i odd 4 1 242.8.a.e 1
260.l odd 4 1 338.8.b.d 2
260.p even 4 1 338.8.a.d 1
260.s odd 4 1 338.8.b.d 2
340.r even 4 1 578.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a ✓ 1 20.e even 4 1
16.8.a.b 1 5.c odd 4 1
18.8.a.b 1 60.l odd 4 1
50.8.a.g 1 20.e even 4 1
50.8.b.c 2 4.b odd 2 1
50.8.b.c 2 20.d odd 2 1
64.8.a.c 1 40.k even 4 1
64.8.a.e 1 40.i odd 4 1
98.8.a.a 1 140.j odd 4 1
98.8.c.d 2 140.w even 12 2
98.8.c.e 2 140.x odd 12 2
144.8.a.i 1 15.e even 4 1
162.8.c.a 2 180.v odd 12 2
162.8.c.l 2 180.x even 12 2
242.8.a.e 1 220.i odd 4 1
256.8.b.b 2 80.j even 4 1
256.8.b.b 2 80.s even 4 1
256.8.b.f 2 80.i odd 4 1
256.8.b.f 2 80.t odd 4 1
338.8.a.d 1 260.p even 4 1
338.8.b.d 2 260.l odd 4 1
338.8.b.d 2 260.s odd 4 1
400.8.a.l 1 5.c odd 4 1
400.8.c.j 2 1.a even 1 1 trivial
400.8.c.j 2 5.b even 2 1 inner
450.8.a.c 1 60.l odd 4 1
450.8.c.g 2 12.b even 2 1
450.8.c.g 2 60.h even 2 1
576.8.a.f 1 120.w even 4 1
576.8.a.g 1 120.q odd 4 1
578.8.a.b 1 340.r even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 144 \) acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 144 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1032256 \) Copy content Toggle raw display
$11$ \( (T + 1092)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1909924 \) Copy content Toggle raw display
$17$ \( T^{2} + 216266436 \) Copy content Toggle raw display
$19$ \( (T + 39940)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4721338944 \) Copy content Toggle raw display
$29$ \( (T - 102570)^{2} \) Copy content Toggle raw display
$31$ \( (T + 227552)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 25768596676 \) Copy content Toggle raw display
$41$ \( (T - 10842)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 397843039504 \) Copy content Toggle raw display
$47$ \( T^{2} + 223403694336 \) Copy content Toggle raw display
$53$ \( T^{2} + 2232089784324 \) Copy content Toggle raw display
$59$ \( (T - 2640660)^{2} \) Copy content Toggle raw display
$61$ \( (T - 827702)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 15877008016 \) Copy content Toggle raw display
$71$ \( (T - 1414728)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 960952799524 \) Copy content Toggle raw display
$79$ \( (T + 3566800)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 32181703643664 \) Copy content Toggle raw display
$89$ \( (T - 11951190)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 75379659165316 \) Copy content Toggle raw display
show more
show less